# Andrei Andreevich Markov

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## Markov, Andrei Andreevich

Born June 2(14), 1856, in Riazan; died July 20, 1922, in Petrograd. Russian mathematician who specialized in number theory, probability theory, and mathematical analysis. An adjunct of the St. Petersburg Academy of Sciences from 1886, a corresponding member from 1890, and an academician from 1896.

The son of a minor official, Markov graduated from the University of St. Petersburg in 1878 with a candidate’s degree and that same year received a gold medal for his paper “On the Integration of Differential Equations With the Aid of Continued Fractions.” He was a privatdocent at the University of St. Petersburg from 1880, a professor from 1886, and a senior professor from 1890.

The themes of Markov’s works were close to those of the older representatives of the St. Petersburg mathematical school, such as P. L. Chebyshev, E. I. Zolctarev, and A. I. Korkin. The brilliant results in number theory obtained by Markov in his master’s dissertation, “On the Binary Quadratic Forms of Positive Determinants” (1880), served as the foundation for further studies in this field. His works in analysis dealt with the theory of continued fractions, the study of limiting values of integrals under various conditions imposed on the integrand, methods of improving the convergence of series, and the theory of best approximations. Markov gave an extraordinarily simple solution to the problem of expressing the least upper bound of the derivative of a polynomial in terms of the least upper bound of the polynomial itself.

In probability theory, Markov filled a gap that had remained in the proof of the central limit theorem and thereby was the first to provide a complete and rigorous proof of the theorem under sufficiently general conditions. Markov’s subsequent papers extended the central limit theorem to sequences of dependent variables and led to the remarkable general scheme of “trials connected in a chain.” On the basis of this elementary scheme, Markov established a number of fundamental laws that laid the foundation for all the modern theories of random Markov processes. Markov worked extensively on various applications of probability theory and, in particular, gave a generally accepted probabilistic foundation for the method of least squares. Markov’s textbook *The Calculus of Probabilities* (1900) exerted a great influence on the development of the science and remains of interest to this day owing to the accuracy of the results and the simple methods used to obtain them. His second textbook, *The Calculus of Finite Differences* (1886, lithograph edition; 2nd ed., 1910), was also widely used.

Markov was a progressive scientist who exposed reactionary trends in science; he also opposed the tsarist government’s refusal to confirm the election of M. Gorky to honorary membership in the Academy of Sciences.

### WORKS

*Izbr. Trudy; Teoriia Chisel; Teoriia veroiatnostei*. Moscow, 1951. (Contains a biography written by A. A. Markov [son] and a bibliography of Markov’s works and of the literature on him.)

*Isbrannye trudy po teorii nepreryvnykh drobei i teorii funktsii, naimenee ukloniaiushchikhsia ot nulia*. Moscow-Leningrad, 1948.

*Ischislenie veroiatnostei*, 4th ed. Moscow, 1924.

### REFERENCE

*Materialy dlia biograficheskogo slovaria deistvitel’nykh chlenov Akademii Nauk*, part 2. St. Petersburg, 1917. (Autobiography and list of Markov’s works.)